31. At what points does the curve r(t) = ti + (2t - 1²) k inter- sect the paraboloid z = x² + y²?
31,43 pl
ans 31.
To determine the points of intersection between the curve r(t) = t i + (2t - t^2) k and the paraboloid z = x^2 + y^2, we need to find the values of t that satisfy both equations.
First, we can substitute the equation for the curve into the equation for the paraboloid to get an equation in terms of t:
z = x^2 + y^2 = (ti)^2 + [(2t - t^2)k]^2 = t^2 + 4t^2 - 4t^3 + t^4
Simplifying this equation, we get:
z = t^4 - 4t^3 + 5t^2
Now we can solve for the values of t that satisfy this equation. We can do this by setting z equal to the expression we just derived and solving for t:
t^4 - 4t^3 + 5t^2 - z = 0
This is a quartic equation, which can be difficult to solve. However, we can make use of the fact that the curve r(t) is a parametric equation of a line in 3D space, so we know that it intersects the paraboloid in at most two points. This means that the quartic equation we just derived has at most two real roots, and we can use this fact to simplify our calculations.
To find the values of t that satisfy the equation, we can use numerical methods such as Newton's method or the bisection method. Alternatively, we can graph the two equations in a 3D coordinate system and visually determine the points of intersection.
In summary, to find the points of intersection between the curve r(t) = t i + (2t - t^2) k and the paraboloid z = x^2 + y^2, we need to substitute the equation for the curve into the equation for the paraboloid to get an equation in terms of t, and then solve for the values of t that satisfy the equation. This can be done using numerical methods or by graphing the two equations in a 3D coordinate system.
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